Implicit operations in varieties of commutative monoids

Abstract

An implicit operation of a class of similar algebras $\mathsf{K}$ is a collection of first order definable partial functions on the members of $\mathsf{K}$ that is globally preserved by homomorphisms. For instance, "taking inverses" can be viewed as a unary implicit operation of the class of all monoids because its graph on a given monoid is defined by the equation $xy \thickapprox 1 \thickapprox yx$ and monoid homomorphisms preserve existing inverses. As this example demonstrates, the implicit operations of a class $\mathsf{K}$ need not be given by a term of $\mathsf{K}$. We show that an equational class of commutative monoids can be expanded with enough implicit operations so that every implicit operation can be interpolated by a family of terms if and only, in each of its members, for every $a$ there exists some $b$ such that $a = a^2b$, i.e., the class consists of inverse monoids. Our methods build on the interaction of the theory of implicit operations with Grillet's description of finitely generated subdirectly irreducible commutative semigroups and the combinatorics deriving from an extension of Isbell's Zigzag Theorem to all equational classes of commutative monoids.